Download Peking University Industrial Organization, Spring 2013 Guanghua

Peking University
Guanghua School of Management
Industrial Organization, Spring 2013
L. Shi
Problem Set 3
Out on March 11, due on Monday, March 18, 2013
Question 1: Cournot Competition
Industry demand, P=100-2Q; Q= q1  q2 ,
Cost for firm1, C1 ( q1 )  4q1 ,
Cost for firm 2, C2 ( q2 )  2q2
(a) Write the two profit functions.
(b) Find the best response functions for the two firms.
(c) Draw the two best responses with firm 2 output on the vertical and firm 1 output
on the horizontal axis. Explain why the intersection of the two best responses
gives the equilibrium output choices.
(d) Compute the equilibrium quantities and profits for the two firms.
(e) What changes if the marginal cost of firm 2 increased to $4 per unit?
(f) Compare the perfectly competitive, monopoly and the Cournot outputs, prices,
and profits when each firm has a marginal cost of $4 per unit.
Question 2: Bertrand Competition
If firm 1 has constant marginal cost c1 and firm 2 has constant marginal cost c2 , with c1 <
c2 , then
(a) show that firm 1 will be the only one selling in the market.
(b) what will be the market price? (hint: it depends on whether p1m is greater or lower
than c2 where p1m is the price firm 1 sets assuming firm 2 is not present)
(c) What are the features that we do not find very realistic in the Bertrand model?
Question 3: Cournot equilibrium, Stackelberg equilibrium
Market demand: P=100-Q, Q= q1  q2
Cost for firm1, C1 ( q1 )  q12 ,
Cost for firm 2, C2 ( q2 )  2q22
(a) if these two firms were Cournot competitors, what would be the equilibrium
output levels, the market price and the profits of the two firms? Compare your
results in this case with the Cartel outcomes (in homework week 2).
(b) if firm 1 is the leader and firm 2 the follower, find the Stackelberg equilibrium
(outputs), the market price, and the profits of the two firms. Compare your results
in this case with the Cournot outcome.
Product Differentiation and Monopolistic Competition
Question 4
All firms in a Cournot monopolistic competitive industry have the same cost function
C(q) = 25 + 10q. Market demand is Q = 110 – p.
(a) Calculate the equilibrium price, firm output, total output and number of firms in the
industry.
(b) What is the first-best outcome, i.e., the outcome that maximizes the total surplus and
its total surplus?
(c) What is the total surplus in the monopolistic competition equilibrium?
(d) How would the monopolistic competition equilibrium change if a franchise tax of $75
were imposed on each firm?
(e) Calculate the equilibrium price, firm output, total output and number of firms in the
industry if a technical innovation were to reduce unit production costs to $5?
Question 5
In models of monopolistic competition, entry of comparable goods makes each
competitor’s residual demand curve retreat until it is just tangent to the average cost
curve. Each competitor then prices at average cost, for a profit of zero. Using a linear
demand curve, p = a – bQ, prove (can be graphically) that there is no contradiction
between having average cost pricing and having marginal cost equal marginal revenue
(the usual profit maximizing condition).
Question 6
In the Bertrand Model with product differentiation, suppose that the two Bertrand firms
face the following symmetric demand curves:
q1  96  2 p1  0.5 p2
q2  96  2 p2  0.5 p1
where q1 , q2  0, p1 , p2  48.
a. Find the Bertrand equilibrium.
b. Is product differentiation more or less significant, and equilibrium price higher or lower
in the above example than in the following example with
q1  96  2 p1  p2
q2  96  2 p2  p1
where q1 , q2  0, p1 , p2  48.